Magneto hydrodynamic Rayleigh Problem with Hall Effect and Rotation in the Presence of Heat Transfer

Hochgeladen von

Beschreibung:

The main objective of the paper is to investigate the magneto hydrodynamic flow version of
the Rayleigh problem including Hall effect and Rotation in the presence of Heat transfer. Analytical
solution has been found depending on the physical parameters including the Hall parameter
N ,
Hartmann
number
M , Grashof number
Gr , Prandtl number
Pr
and the Rotation parameter
2 K
. The influence of
these parameters on velocity profiles and temperature profiles are demonstrated graphically and the results
are discussed. Further, it is observed that increase in the Hall parameter and Rotation parameter leads to
decrease in the velocity profiles. It is also found that the increase of Grashof number results in the decrease
of primary flow and increase of secondary flows.

Magneto hydrodynamic Rayleigh Problem with Hall Effect and Rotation in the Presence of Heat Transfer

Hochgeladen von

Beschreibung:

The main objective of the paper is to investigate the magneto hydrodynamic flow version of
the Rayleigh problem including Hall effect and Rotation in the presence of Heat transfer. Analytical
solution has been found depending on the physical parameters including the Hall parameter
N ,
Hartmann
number
M , Grashof number
Gr , Prandtl number
Pr
and the Rotation parameter
2 K
. The influence of
these parameters on velocity profiles and temperature profiles are demonstrated graphically and the results
are discussed. Further, it is observed that increase in the Hall parameter and Rotation parameter leads to
decrease in the velocity profiles. It is also found that the increase of Grashof number results in the decrease
of primary flow and increase of secondary flows.

ABSTRACT: The main objective of the paper is to investigate the magneto hydrodynamic flow version ofthe Rayleigh problem including Hall effect and Rotation in the presence of Heat transfer. Analyticalsolution has been found depending on the physical parameters including the Hall parameter N , Hartmannnumber M , Grashof number Gr , Prandtl number Pr and the Rotation parameter K 2 . The influence ofthese parameters on velocity profiles and temperature profiles are demonstrated graphically and the resultsare discussed. Further, it is observed that increase in the Hall parameter and Rotation parameter leads todecrease in the velocity profiles. It is also found that the increase of Grashof number results in the decreaseof primary flow and increase of secondary flows.

Keywords:- MHD flow, Hall Effect, viscous fluid, Heat transfer, RotationI. INTRODUCTIONIn recent years, the analysis of hydromagnetic flow has applications in diverse fields of Science andTechnology such as soil sciences, astrophysics, nuclear power reactors etc. The study of MHD flow problemshas achieved remarkable interest due to its application in MHD generators, MHD pumps and MHD flow metersetc. Geophysics encounters MHD phenomena in interaction on conducting fluids and magnetic fluids. Therotating flow of an electrically conducting fluid in presence of magnetic field has got its importance inGeophysical problems. The study of rotating flow problems is also important in the solar physics dealing withthe sunspot development, the solar cycle and the structure of rotating magnetic stars.Stokes analysed the flow of an incompressible viscous fluid past an impulsively started infinitehorizontal plate in its own plane. Rossow[1] initiated Rayleighs problem for non-conducting plate whileChang&Yen[2] have taken the plate to be perfectly conducting in a various transverse magnetic field. The effectof coriolis forces on Rayleigh Problem was considered by Sathi [3]. Rayleigh problem in MHD with suctionwas considered by Girish Chandra Pandey[4].The Hall effect is due merely to the sideways magnetic force on the drifting free charges. The electricfield has to have a component transverse to the direction of the current density to balance this force. In manyworks on plasma physics, the Hall effect cannot be ignored as it has a significant effect on the flow pattern of anionized gas. Hall effect results in a development of an additional potential difference between opposite surfacesof a conductor for which a current is induced perpendicular to both the electric and magnetic field. This currentis termed as Hall current. The effect of Hall current on MHD Rayleighs problem in ionized gas was discussedby Mohanty[5].Solution of Rayleigh problem for conducting fluid was considered by Abd-el-Malek etal[6].Magneto hydrodynamic Rayleigh problem with Hall effect was studied by Haytham Sulieman[7]. Deivanayakietal[8] studied the MHD Rayleigh problem with Hall effect and Rotation. In this study the effect of the Hallcurrent and rotation on the magneto hydrodynamic flow version of the classical Rayleigh problem in thepresence of Heat transfer was considered.

II. FORMULATION OF THE PROBLEM

The scenario under investigation comprises an incompressible electrically conducting, viscous fluidpast an infinite vertical plate occupying the plane y = 0. The -axis is taken in the direction of the motion of theplate and axis lying on the plate normal to both and y axis. Initially it is assumed that the plate and thefluid rotate in unison with a uniform angular velocity about the y - axis normal to the plane are at the sametemperature everywhere in the fluid. At time > 0, the plate starts moving impulsively with the uniformvelocity in its own plane along the -axis. Also the temperature of the plate is raised/lowered to . A uniformmagnetic field 0 , parallel to - axis is imposed. The schematic diagram is given by| IJMER | ISSN: 22496645 |

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| Vol. 5 | Iss. 8 | August 2015 | 59 |

Magneto hydrodynamic Rayleigh Problem with Hall Effect and Rotation in the Presence of Heat Transfer

Fig 1: Schematic representation

Under the above assumptions, in the absence of an input electric field the governing boundary layer equationsareEquations of continuity . =0(2.1)Equation of motion

11+ . q + 2 q = P + 2 q + J H + g(2.2)

The energy equation

(is the electrical conductivity). Here is the current density, t is the time, is density, is

kinematic viscosity, e is electric charge, m is mass of an electron, n is the electron number density, is the meancollision time and is magnetic permeability.Let = (, )The initial and boundary conditions are 0: = 0, = 0, = 0 for 0 > 0: = 0 , = 0, 0, = for = 0 = 0, = 0, , = 0 as 0, = 0 , 0 as (2.5)At infinity the magnetic induction is uniform with components (0,H,0), and hence the current density vanishesand since the free stream is at rest, it follows from generalized Ohms law that = 0 as . Assuming smallmagnetic Reynolds number for the flow, the induced magnetic field is neglected in comparison to the appliedconstant field 0 .Introducing the non-dimensional quantities: = =

0 .

, =

, 2 =

, =

02

2 , =

, =

02

03

, = , =

, = , 2 =

02 ,02

(2.6)

All the physical variables are defined in the Nomenclature.

Equations (2.1), (2.2) and (2.3) transform to the following non-dimensional forms, respectively.

Also substitute , = () in (3.1), we have

+ = The equation (3.7) can be solved under the boundary conditions, 0 = , = 0Therefore the solution is = +

(+ )

(+ )

= 3 4 + 3 4 + 2 + 2

1 1 2 + 2

4 2 + 2

1 1 2 + 2

(3.9) 3 4 2 + 2

1 1

2 + 2

= 3 4 3 4 3

(3.8)

Separating equation (3.9) into real and imaginary parts, we have

3 4

(3.7)

1 2 + 2

(3.10)

3 4 2 + 2

(3.11)

The constants involved in the above discussion have been obtained but not presented here for the sake ofbrevity.In order to attain a physical insight into the problem, we have carried out numerical calculations for the velocityfields and temperature field at the plate due to time, Hartmann number, Hall parameter, Rotation parameter,Prandtl

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| Vol. 5 | Iss. 8 | August 2015 | 61 |

Magneto hydrodynamic Rayleigh Problem with Hall Effect and Rotation in the Presence of Heat Transfer

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| Vol. 5 | Iss. 8 | August 2015 | 62 |

Magneto hydrodynamic Rayleigh Problem with Hall Effect and Rotation in the Presence of Heat Transfer

IV.

CONCLUSION

Fig. 2, 4 and 6 depict the variation of the velocity component against under the influence ofHartmann number, Hall parameter and Grashof number of heat transfer respectively. The velocity componentdecreases in all the cases. From fig. 3 and 7 it is clear that the velocity component increases against underthe influence of Hartmann number and Grashof number. In fig. 5, the effect of Hall parameter on the velocitycomponent against are shown and decreases.We see from fig. 8 and 9 that increase and decrease when the effect of Prandtl number isconsidered. Fig.10 and 11 shows that the velocity components and decreases with the increase of Rotationparameter against . The temperature profile decreases with the increase of time against as shown in Fig.12.